[EM] Fw: borda count

[Warren Schudy]

On Sun, 7 Nov 2004, Steve Eppley wrote:
> If James used the term Borda to refer to any scoring rule, 
> which is a possible interpretation of his premise about 
> "no matter how" the points are allocated to successive 
> preferences, then his supposition is wrong.  Plurality 
> rule is a scoring rule where S1 = 1 and S2=S3=...=Sn=0,
> and it always elects a candidate ranked first by an 
> absolute majority, if such a candidate exists.  Any scoring 
> rule such that S1 >= 2 * (S2 + S3 + ... + Sn) has that 
> property. (So do the scoring rules where n = 2 and 
> S1 > S2.)

Maybe I shouldn't beat a dead horse (voting method?), but that condition
on scoring rules electing a 1st-by-majority is wrong. As a 
counter-example, consider 3 candidates with S1=2,S2=1,S3=0. This satisfies 
the condition on S1 >= 2*(S2+...Sn), but with the following ballots it 
doesn't elect the 1st-choice majority:

51: A>B>C
49: B>C>A

A: 102 B: 149 C: 49

Another way to see that its wrong: scoring rules produce the same result 
if one replaces S_i by M*S_i+L, M>0, but S1>=2*(S2+...SN) changes when 
such a transformation is made to the S_i.

I think that plurality is the only scoring rule that also elects a 
candidate first-choice by a majority. Consider S1=1, S2=s, S3=0, and 
fraction 0.5+f voting A>B>C, 0.5-f voting B>C>A.

A: 0.5+f
B: 0.5-f+s*(f+0.5)

A-B = 2f-s*(f+0.5)

If f < 1/(4/s-2), then A loses despite getting more than half the votes. 
This shows that there is no scoring rule that ensures this form of 
majority rule for the N=3 case other than plurality. (Replacing S_i with 
L*S_i+M for L,M independant of i, L>0, does not affect the outcome, so 
setting S1=1,Sn=0 does not lose any generality other than the silly rule 
S1=S2=S3)

For N>3, the same result holds: use the same example, with C replaced by 
N-2 clones of C.

-wjs

/-----------------------------------------\
| Warren Schudy                           |
| WPI Class of 2005                       |
| Physics and computer science major      |
| AIM: WJSchudy  email: wschudy at wpi.edu   |
| http://users.wpi.edu/~wschudy/          |
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